Algebra 1 Math Review

x coordinate of vertex

-b/2a

transformation

A movement of a geometric figure

alternate exterior angles

Angles that lay outside the parallel lines and are on opposite sides of the transversal; They are congruent.

mean

Average

area

Length x Width

circumference

The distance around a circle

Power of a Power

multiply the exponents

Closure under multiplication and addition

they produce real numbers

subtract exponents

to divide powers having the same base,

Transforming absolute value inequalities

(expression)<c means -c<expression<c (expression)>c means expression>c or expression<-c

linear function

A function whose graph is a straight line.

constant

A value that does not change

area of a triangle

A=1/2bh

Addition Property of Equality

If x=y, then x+z=y+z

rotation

The action of rotating around an axis or center.

image

after the transformation

Pythagorean Theorem

a²+b²=c²

Parabola

the graph of a quadratic function

remote interior angles

the two nonadjacent interior angles corresponding to each exterior angle of a triangle

slope of a vertical line

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Multiplication Property of -1

-1*x=-x

slope of a horizontal line

0

Converse of the Multiplication Property of Zero

If a product of real numbers equals zero, then one of its factors equals zero. That is, for real numbers n and p, if np =0, then n=0 or p=0

Addition Property of Order

If a<b, then a+c<b+c

Square Root Property of Equality

If a=b, then square root of a= square root of b

mode

Most

volume of cone

V=1/3πr²h

perfect square

a square of a whole number

Associative Axiom for Addition

(x+y)+z=x+(y+z)

Like terms

Expressions that contain the same variables to the same power

negative exponents

For any nonzero number a^-n is the reciprocal of a^n

volume of triangular prism

V=(1/2bh)h

abscissa

x-axis

Associative Axiom for Multiplication

(xy)z=x(yz)

completing the square

1. Take half of the coefficient of X. 2. Square it. 3. add the result.

simple radical form

1. There are no radicals in the denominator 2. The radicand(s) in the numerator are as small as possible

Splitting the middle term

1. multiply a by c 2. Look for two factors of ac whose sum is b 3. split the middle term and factor by grouping (If not factors then it is prime)

Three Major Things Done in Algebra

1. write an expression representing something in the real world 2. find the value of an expression when you know what x equals. 3. find the value of x when you know what the expression equals.

Names for large numbers

10^3= thousand 10^6= million 10^9= billion 10^12= trillion 10^15= quadrillion 10^18= quintillion

Quadrants of Cartesian Grid

2,1,3,4

Product of Conjugate Radical Binomials

=x-y, which is a rational number

Graphs of Quadratic Inequalities

> then outside endpoints (or all of them). If < then the graph is between the endpoints

discriminant test for factorability

A Quadratic trinomial can be factored if and only if the discriminant is a perfect square

negative correlation

A finding that two factors vary systematically in opposite directions, one increasing as the other decreases.

non-linear function

A function whose graph is not a straight line

scatter plot

A graph used to determine whether there is a relationship between paired data

scatter plot

A graph with points plotted to show a possible relationship between two sets of data.

exponent

A mathematical notation indicating the number of times a quantity is multiplied by itself.

slope

A measure of the steepness of a line. Given two points with coordinates (X1,Y1) and (x2,y2) on a line the slope, m, of the line is given m = rise/run= y2-y1/x2-x1

scientific notation

A method of writing or displaying numbers in terms of a decimal number between 1 and 10 multiplied by a power of 10.

composite number

A number greater than 0 that has more than two different factors.

coefficient

A number multiplied by a variable in an algebraic expression.

coefficient

A number multiplied by a variable.

Rational Number

A number that can be written as a fraction

rational number

A number that can be written as a fraction

Irrational Number

A number that cannot be written as a quotient of two integers.

corresponding angles

A pair of angles which are on the same side of the transversal, one must be interior, one must be exterior, and they must be nonadjacent

ordered pair

A pair of numbers that can be used to locate a point on a coordinate plane

ordered pair

A pair of numbers that give the location of a point on a x and y grid. for example graph (3, 2)

Vertex

A point where two or more lines meet.

vertex

A point where two or more straight lines meet.

function

A prewritten formula that makes it easy to perform common calculations

Radical Equation

A radical equation is an equation with a variable under the radical sign.

system of equations

A set of equations with the same variables

inequality

A statement that compares two quantities using <, >, ≤,≥, or ≠

radius

A straight line from the center to the circumference of a circle or sphere.

diameter

A straight line passing from side to side through the center of a circle or sphere.

reflection

A transformation that "flips" a figure over a mirror or reflection line.

translation

A transformation that "slides" each point of a figure the same distance in the same direction.

linear relationship

A type of relationship that exists between two variables whose graphed data points lie on a straight line

area of a rectangle

A=lw

area of a circle

A=pie r squared

Subtraction

Adding a number's opposite

real number

All rational and irrational numbers

Integer

All whole numbers (both positive and negative) and zero.

integer

All whole numbers (both positive and negative) and zero.

obtuse angle

An angle between 90 and 180 degrees

acute angle

An angle that is less than 90 degrees

identity

An equation that is true for all values of the variable.

irrational number

Any real number which cant be expressed as a fraction of two integers

Exponentiation

Arithmetic operator that raises to a power

standard form

Ax+By=C

associative property

Changing the grouping of numbers will NOT change the value. For example: (7 + 4) + 8 = 7 + (4 + 8) also works with multiplication

output

Data that has been processed into a useful format.

input

Data that is to be entered into a worksheet

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Denominator equals zero

quadrants

Four regions into which a coordinate plane is divided by the x-axis and the y-axis

Statistics

Given data resulting from a random experiment, predict something about the real-world situation from which the data comes.

Probability

Given information about something in the real world, predict the results of a random experiment.

congruent

Having the same size and shape

Common factor

If in an expression each term has the same factor.

Multiplication Property of Order

If x<y then xz<yz if z is positive xz>yz if z is negative xz=yz if z is zero

Quotient of two powers with equal bases

If you divide two powers, and they have equal bases, then you can subtract the exponents and use that same base.

pythagorean theorem

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs

interior angles

Inside angles of two lines intersected by a transversal

Prime

Is a positive integer that has no positive integers as factors other than 1 and itself. The number 1 is not considered to be prime.

Division

Multiplying by its reciprocal, besides zero.

negative exponents

Negative exponents always have to SWITCH positions in their fractions. [If you don't see a fraction, make one by drawing a bar over 1, then switch 3^(-1)=1/3

Probability Equation

Number of favorable outcomes/total number of outcomes= Number of elements in the event space/ number of elements in the sample space

Terms

Numbers that are added to each other or subtracted from each other.

Factors

Numbers that are multiplied together.

Order of Operations

Parentheses, powers, multiply and divide from left to right, and add and subtract from left to right.

negative exponents

Raising a number to a negative exponent is the same as raising the number's reciprocal to the equivalent positive exponent

slope

Rise over run

order of operations

Rules describing what sequence to use in evaluating expressions

slopes of perpendicular lines

Slopes are negative reciprocals or each other

Square Root of a Quotient

Square root distributes over division

Square root of a product

Square root distributes over multiplication.

Power, Base, and Exponent

The base is the number on the bottom, the exponent is the number on top, and together they form a power.

absolute value

The distance a number is from zero on a number line. ALWAYS POSITIVE

perimeter

The distance around a figure

distance formula

The distance d between any two points with coordinates

Domain

The domain of a function is the set of permissible values of the independent variable.

independent variable

The experimental factor that is manipulated; the variable whose effect is being studied.

Characteristic

The factor on the right power of ten

Intersection

The intersection of sets A and B, abbreviated A Upside down U B, is the set of numbers that are in both sets

least common multiple (LCM)

The least number that is a common multiple of two or more numbers.

median

The middle number in the set (if two numbers are in the middle, find the average of the two numbers)

nth root

The number that must be multiplied times itself n times to equal a given value.

dependent variable

The outcome factor; the variable that may change in response to manipulations of the independent variable.

commutative property

The property that says that two or more numbers can be added or multiplied in any order without changing the result.

Range

The range of a function is the set of values of the dependent variable corresponding to all values of the independent variable in the domain.

Sign of a Product

The sign of a product will be: NEGATIVE, if it has an ODD number of negative factors. POSITIVE, if it has an EVEN number of negative factors.

Union

The union of sets A and B, abbreviated A U B, is the set of numbers that are either in set A or in set B

no correlation

There is no relationship between data sets.

Relatively prime

Two numbers are relatively prime if their greatest common factor is 1

Additive Inverses or Opposites

Two numbers whose sum is 0 are additive inverses of one another.

volume of rectangular prism

V=lwh

exterior angles

When the sides of a triangle are extended, the angles that are adjacent to the interior angles.

repeating decimal

a decimal that repeats a digit or group of digits forever

Property

a fact that is true concerning the system.

Event

a favorable outcome of a random experiment

linear function

a function in which the graph of the solutions forms a line

Function

a function is a set of ordered pairs for which there is never more than one value of y for any one given value of x

cluster

a grouping of a number of similar things

transversal

a line that intersects two or more coplanar lines at different points

scientific notation

a method of writing very large (34,200,000) or very small (0.0000029) numbers using powers of 10

rational number

a number that can be expressed as a ratio (fraction) of two integers - terminating and repeating decimals are considered rational

Common multiple

a number that is a multiple of two or more numbers

square root

a number that when multiplied by itself equals a given number

axiom

a property that forms the base of the mathematical system. It is assumed to be true without proof.

System of Equations

a set of equations with the same variables

Linear Function

a set of ordered pairs that are related by an equation of the form

Pythagorean theorem

a squared+ b squared= c squared

radical sign

a symbol to indicate the square root

dilation

a transformation created proportionally by enlarging or reducing a figure ( multiply x and y by scale factor)

the sign above the radical sign

a viniculum

distributive property

a(b + c) = ab + ac

Random Experiment

an action in which there is no way to tell beforehand just which of several possible things will happen

linear regression

an algorithm to find a precise line of fit for a set of data

fractional equation

an equation in which a variable occurs in a denominator

conditional equation

an equation that has one or more answers that may or may not satisfy the equation

linear equation

an equation whose graph is a straight line

Open Sentence

an equation with one or more variables

Solution of a system

an ordered pair that satisfies all equations in the system

alternate interior angles

angles are on opposite sides of the transversal and on the inside of the given lines

zero exponent

any nonzero number raised to the zero power is 1

Pythagorean theorem

a²+b²=c²

The Discriminant

b squared minus 4ac

pre-image

before the transformation

conjugate binomials

binomials that are the same except for the sign between the terms

run

change in x

rise

change in y

expressions

consist of numbers, variables and operations

identity property

describes the results of adding the number zero to another number or multiplying a number by one - the results are identical

Fundamental Theorem of Arithmetic

each positive integer has exactly one set of prime factors

Outcomes

equally likely results of a random experiment

Power of a Quotient

exponentiation distributes over division

Power of a Product

exponentiation distributes over multiplication.

First step in factoring

factor the GCF of the terms from the polynomial

slopes of parallel lines

have to have same slope

Square root Property of order

if a>b then the square of a > than the square of b

Transitive Axiom of Equality

if x=y and y=z then x=z

Symmetric Axiom of Equality

if x=y then y=x

Multiplication Property of Equality

if x=y, then xz=yz

equations of a horizontal and vertical lines

if y is constant, the graph is a horizontal line. If x is constant, than the graph is a vertical line.

Cross multiplication

in a proportion, the product of the means equals the product of the extremes. a:b= d:c then ad=bc

Polynomial

is an expression that has no operation other than addition, subtraction, and multiplication by or of the variable (s).

regression line

line of best fit

parallel lines

lines in the same plane that never intersect

Polynomials with one, two, or three terms

monomial, binomial, and trinomial

add exponents

multiplying powers with the same base

Exponent Property of the nth root

n over the square root x= x and one nth

consecutive

numbers that follow one after the other in a sequence (1, 2, 3...x, x+1, X+2)

legs

sides that are adjacent to the right angle

180 degree rotation

switch sign of x and y to the opposite sign.

90 degree rotation

switch x and y location and use sign (-, +) to the quadrant the point will be after rotation.

radical

taking the square root of a number

Degree

the degree of a polynomial with one variable is the exponent of the highest power of that variable. Linear, Quadratic, and Cubic.

difference of two squares

the factors of a difference of two square are conjugate binomials.

least common denominator

the least common multiple of two or more denominators

hypotenuse

the longest side of a right triangle

line of best fit

the most accurate trend line on a scatter plot showing the relationship between two sets of data

significant digits

the number of significant digits is the number in the mantissa. The answer is put with the number with the least amount of significant digits.

Numerical Coefficient

the number that stands before a variable

radicand

the number you're taking the square root of

Mantissa

the power on the left side of the decimal point after the first number other than zero in an exponent

rational algebraic expression

the quotient of two polynomials

scale factor

the ratio used to enlarge or reduce similar figures

positive correlation

the relationship between two sets of data in which both sets of data increase together.

direct of variation

the relationship of two variables if there is a nonzero number

event space

the set of all favorable outcomes for an event

Sample Space

the set of all possible outcomes for a random experiment.

Real numbers

the set of all rational and irrational numbers

radical symbol

the symbol used to denote a root

Factoring by grouping

the use of the distributive property to factor some polynomials having four or more terms

volume of sphere

the volume of a sphere equals four thirds times pies times r to the third power

Canceling

to divide the numerator and denominator by the same common factor.

Multiplication property of fractions

to multiply two fractions, multiply their numerators and multiply their denominators

average speed equation

total distance/total time

system of linear equations

two or more linear equations.

nonlinear relationship

whenever one quantity is plotted with another and the resulting graph is NOT a straight line.

x intercept

where the graph crosses the x-axis

y-intercept

where the line crosses the y axis

Multiplicative Identity Axiom

x * 1 = x

Multiplicative Inverses Axiom

x * 1/x = 1

Additive Inverses Axiom

x + (-x) = 0

Factor Theorem

x -n is a factor of a polynomial if and only if the polynomial equals to 0 when n is substituted for x

Quadratic Formula

x equals the opposite of b plus or minus the square root of b square minus four a c all divided by 2a

Between and inclusive

x is between means: a<x<b x is between, inclusive means: a<(with line)x<(with line) b

Distributive axiom for multiplication over addition

x(y+z) = xy+ xz

Multiplication Property of Zero

x*0=0

Additive Identity Axiom

x+0=x

Commutative axiom for addition

x+y=y+x

Special Cases of fractions

x-5/5-x=-1, x+5/x+5=1, and nothing is special about conjugate numbers

multiplication identity axiom

x.1=x

Reflexive Axiom of Equality

x=x

Commutative axiom for multiplication

xy=yx

Ordinate

y-axis

ordinate

y-axis

point-slope form

y-y1 = m(x-x1)

Point slope form of a linear equation

y-y1=m(x-x1)

point slope form

y-y1=m(x-x1)

Quadratic Function

y=axsquared+ bx+ c

Inverse Variation Function

y=k/x

Direct Variation Function

y=kx

General Equation vs. Particular equation

y=kx or y=k/x is a general equation and y=3x or y=24/x is a particular equation

slope intercept form of an equation

y=mx + b (m is the slope, b is the y intercept point)

regression equation

y=mx+b

slope-intercept form

y=mx+b

product of two powers with equal bases

you can add the exponents